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ω
A cylinder of diameter d floats with l of its length submerged. The total height is L. Assume no damping. At time t = 0 the cylinder is pushed down a distance B and released.
What is the frequency of oscillation?
[itex] f = ω/2\pi [/itex]
[itex] Ma = F_{(bouyancy)} [/itex]
Writing this in our differential form, making proper substitutions, and noting that bouyancy is affected by the distance that our cylinder is submerged we come to.
[itex] dx^2 (M_{(cylinder)}) + x (\rho_{(water)} g Area_{(cylinder face)})= 0 [/itex]
we know that [itex]M_{(cylinder)} = V_{(cylinder)}\rho_{(cylinder)} [/itex]
hence we should have
[itex] ω^2 = (\rho_{(water)} g Area_{(cylinder face)}) / V_{cylinder}\rho_{(cylinder)} = g\rho_{(water)} / l \rho_{(cylinder)}[/itex]
however the solution in my problem set has ω^2 = g/l. Can anyone shed some light on why the densities may cancel??
Homework Statement
A cylinder of diameter d floats with l of its length submerged. The total height is L. Assume no damping. At time t = 0 the cylinder is pushed down a distance B and released.
What is the frequency of oscillation?
Homework Equations
[itex] f = ω/2\pi [/itex]
[itex] Ma = F_{(bouyancy)} [/itex]
Writing this in our differential form, making proper substitutions, and noting that bouyancy is affected by the distance that our cylinder is submerged we come to.
[itex] dx^2 (M_{(cylinder)}) + x (\rho_{(water)} g Area_{(cylinder face)})= 0 [/itex]
we know that [itex]M_{(cylinder)} = V_{(cylinder)}\rho_{(cylinder)} [/itex]
hence we should have
[itex] ω^2 = (\rho_{(water)} g Area_{(cylinder face)}) / V_{cylinder}\rho_{(cylinder)} = g\rho_{(water)} / l \rho_{(cylinder)}[/itex]
however the solution in my problem set has ω^2 = g/l. Can anyone shed some light on why the densities may cancel??